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Autocracy, Devolution and Growth
Elena Paltseva
January 2007
Abstract
Some autocracies have sustained high economic growth for many decades;others have stagnated at low levels of production. Paradoxically, the stagnat-ing autocracies appear to possess more natural resources and be more resistantto political change than the growing autocracies. The paper proposes a newexplanation for these observations. Consider an economy where an innitelylived autocrat determines taxes and the political regime. As capital accumu-lates and growth rates decline, the autocrat faces an increasing temptation toexpropriate the capitalists. Since expropriation eliminates growth, the autocratmay voluntarily refrain from expropriating if future growth is su¢ ciently large;otherwise, the temptation to expropriate can only be resisted through a crediblecommitment, that is, by devolving some political power. For autocrats withlarge benets of control, for example valuable natural resource rents, devolutionof power may always be unattractive. As a result, capitalists realize that theywill eventually be expropriated, and capital accumulation therefore never starts.On the other hand, autocrats with small resource rents will eventually devolvepower, since this commitment is necessary to sustain growth. Therefore, capi-talists are willing to start accumulating despite the autocratic regime. In otherwords, autocracies are vulnerable to the resource curse.
JEL classication: D02, D72, D9, O1Keywords: Political Economy, Institutions, Growth, Autocracy, Commit-
ment, Devolution of power, Resource Curse
Stockholm School of Economics and Institute for International Economic Studies, Sweden,[email protected] am indebted to Tore Ellingsen for his advice and encouragement. I also thank Philippe Aghion,Erik Berglöf, Mike Burkart, Chloe Le Coq, Nicola Gennaioli, Sergei Guriev, John Hassler, Ethan Ka-plan, Assar Lindbeck, Elena Panova, Stephen Parente, Torsten Persson, Maria Petrova, Jesper Roine,Konstantin Sonin, Jörgen Weibull and Fabrizio Zilibotti for valuable comments, as well as seminarparticipants at CEFIR, Stockholm School of Economics, IIES and participants at EEA Congress2006 (Vienna). Jan Wallanders and Tom Hedelius Research Foundation is gratefully acknowledgedfor nancial support. All remaining errors are my own.
1
Autocracy, Devolution and Growth 2
1 Introduction
Nowadays, most economists agree that economic and political changes are intertwined.1
For example, it is commonly argued that both protection of property rights from gov-
ernmental abuse creates economic growth, and that economic growth gives rise to
political freedom, constraining the discretion of the executive.2 Still, the relation-
ship between limited government and economic outcomes is not very well understood.
Notably, autocracies are found to be both the best and worst performers in terms
of growth rates.3 Moreover, the most economically successful autocracies tend to
eventually be replaced by more democratic institutions, whereas poorly performing
autocracies often prevail for a very long time.
In this paper, I argue that the scope for capital accumulation and growth in an
autocracy is largely determined by the autocrats incentive to cling to power. A main
result of the model is that there can be private capital accumulation only if the auto-
crats benets from political control are not too large. The reason is that eventually, as
capital accumulates and growth rates decline, an unfettered autocrats temptation to
expropriate capital becomes irresistible. Therefore, capital accumulation proceeds be-
yond this point only after the autocrat has relinquished some power. While autocrats
with small benets of political control are willing to relinquish power once it becomes
necessary in order to sustain future capital accumulation, autocrats with large benets
of control are not. In the latter case, capitalists, looking ahead, realize that they will
eventually be expropriated and never start accumulating.
Natural resource availability might be the most obvious determinant of an auto-
crats incentive to cling to power. If a country is su¢ ciently rich in natural resources,
an autocratic ruler will always resist political change, because of the lost stream of
revenues. In a resource poor country, on the other hand, the autocrat may prefer
sustaining capital accumulation to preserving resource rents. The model can therefore
account both for the observed negative relationship between natural resource rents
and economic growth and the great heterogeneity in economic outcomes under autoc-
racy. It also suggests why resistance to political change may be greater in autocracies
that never experienced capital accumulation than in the economically more successful
autocracies.
The relationship between economic growth and the autocratic rule with its preda-
1E.g. "At some level, these rich dynamics of economic and political change have to be connected",(Persson and Tabellini (2006b)), or "...wealth, its distribution, and the institutions that allocatefactors and distribute incomes are mutually interdependent and evolve together." (Przeworski (2004)).
2The expansion of economic freedom will bring in train greater political freedoms.(Friedman(2002)).
3E.g. Almeida and Ferreira (2002).
1. Introduction 3
tory potential has been (and still is) a topic of an extensive debate. While it is well
recognized that expropriation or other property rights violations by political rulers are
detrimental for investment and growth, there is no consensus on whether the respective
protection of investors has to be institutionalized. According to one view, the abuse
of political power can only be e¤ectively prevented through explicit legal limitations
on rulersauthority. Institutional checks and balances work as a commitment device
against expropriation, which encourages investment and gets growth started. In the
modern literature, this view is associated with works by North and Thomas (1970),
North (1971), North and Weingast (1989) and has inspired a considerable amount of
supportive empirical work.4
An alternative view is that sustained growth can be in the interest of an uncon-
strained dictator, who therefore rationally refrains from expropriation. As emphasized
by Olson (1993), this argument does not require any benevolence on the part of the au-
tocratic ruler; a "stationary bandit" will promote growth for selsh reasons. This view
is also buttressed by the evidence indeed, the most impressive growth episodes were
almost always observed in autocratic regimes.5 Furthermore, Glaeser et. al. (2004)
stress that "initial levels of constraints on the executive do not predict subsequent
economic growth" and "growth (. . . ) may be feasible without immediate institutional
improvement".
Nonetheless, the "stationary bandit" approach does not explain why most of the
well-performing autocracies eventually limit the dictators power, e.g. through par-
tial or full democratization. The classical argument for devolution of power the
famous Lipset (1960) hypothesis suggests that prosperity improves political institu-
tions through better education and the increased importance of the middle class.
In this paper, we complement Lipsets demand-driven view of democratization and
instead look at the supply side of political change. More precisely, we show that a
self-interested dictator may choose to relinquish power in a phase of declining growth.
We thus suggest a compromise between the view of North and Weingast (1989) and
the "stationary bandits" argument of Olson (1993). Olsons argument implies that the
interests of the dictator and the investors are aligned and no commitment device is
needed at all. On the contrary, North and Weingast argue that the incentives of the
ruler are always opposed to those of private investors, so in order to initiate investment
and growth, the autocrat has to commit beforehand. In our model, the conict of
interests between the autocrat and the investors intensies over time, thereby causing
4E.g. Knack and Keefer (1995), Goldsmith (1995), Hall and Jones (1999), Henisz (2000), Keefer(2004).
5E.g. Almeida and Ferreira (2002).
Autocracy, Devolution and Growth 4
a potential delay in the devolution of power.
Specically, we assume that a country is facing an exogenous opportunity for multi-
period investment that is only exploitable by the private sector. This investment is
characterized by decreasing returns to scale. An autocratic ruler can tax the private
sector or expropriate it, but is unable to invest and thus support growth after expro-
priation. The ruler can also choose to relinquish some of her power, which reduces her
payo¤ and deprives her of the right to expropriate. While being in power, the ruler
enjoys private benets of control, (part of) which are no longer available to the ruler
after devolution. That is, the ruler cannot "ne tune" the devolution decision so as
to keep every possible fraction of current economic privileges. The private sector has
an option to avoid taxation and expropriation by diverting resources to a less e¢ cient
alternative use. Due to decreasing returns to scale, the growth rate in this economy
declines over time. As a result, the degree of alignment of governments and private
agentsinterests also decreases: At the early stages of growth, when the growth rate
is high, an autocrat has no incentive to seize assets, because delayed expropriation
signicantly increases the value at stake. As growth slows down, immediate expropri-
ation ("getting the entire pie") becomes increasingly attractive, as compared to the
option of postponing it to increase the size of the pie. The private sector recognizes
these incentives of the ruler, and diverts resources once the ruler is tempted to ex-
propriate. If the rulers private benet of control is relatively low, she commits to
non-expropriation through devolution of power and thereby keeps the private sector
investing. If the private benets of control are high, the ruler does not want to ever
lose them through devolution. Realizing that capital will eventually be expropriated,
the private sector thus never starts to invest. Therefore, the model generates a variety
of development trajectories: The country can face an early limitation of the rulers
power and grow in a non-autocratic regime. Alternatively, the country can grow un-
der maintained autocracy and experience delayed devolution of power. Finally, the
country can stagnate under an autocratic rule with neither devolution nor growth ever
taking place.
The key feature of our model is that devolution of power is a commitment device
against expropriation. A related strand of literature takes devolution of power as a
commitment to redistribution. In Acemoglu and Robinson (2000, 2001), the rich elite
relinquishes the power (by extending franchise) in case it cannot prevent social unrest
through a temporary increase in taxes. Franchise extension transfers the taxation
decision to the (poor) median voter, thereby working as a commitment device for more
redistribution. Gradstein (2004) treats franchise extension as a commitment to private
property rights. In his model, agents allocate their time between production and rent-
1. Introduction 5
seeking, with wealthier agents having a comparative advantage in the latter. With
an income-based franchise, voting participation increases as the economy grows. The
median voter becomes relatively more interested in curbing rent-seeking and protecting
private property rights. Commitment to protect property rights in the future can thus
be achieved by reduction in the franchise threshold. While these approaches provide
an alternative explanation for devolution of power, they ignore the predatory nature
of autocratic rulers, which is key to our argument.
Bates et. al. (2005) do consider predatory rulers. In their model, a government
can be benevolent or opportunistic, but its type is unobservable to the citizens ex
ante. The opportunistic government may decide to show restrains to pretend that
it is benevolent in order to stimulate private investment and increase the appropri-
able assets in the future. In this model, the economy reaches full development under
autocracy in case of a benevolent government and collapses in case of a predatory gov-
ernment. Robinson (2001) associates predatory behavior with ine¢ ciently low public
investment and suggests that the development-enhancing policies may facilitate the
political power contest. Therefore, the party in power may refrain from promoting
development. However, these models do not address the issue of the devolution of
power, which is a fundamental aspect of our framework.
The model predicts that the economies stagnating under an autocratic rule are
characterized by high private benets of control, e.g. are abundant with appropriable
natural resources. This nding parallels the "resource curse" literature, arguing that
the natural resource wealth can be detrimental to countrieseconomic development.
The early work on the "resource curse" attributes the underperformance of resource-
rich countries to economic factors such as "Dutch disease" or deteriorating terms of
trade for the primary commodities. Empirically, this e¤ect is documented e.g. by
Sachs and Warner (1995) and Gylfason (2000). More recent literature emphasizes the
political and institutional determinants of the resource curse (see e.g. Robinson et al.
(2006), Mehlum et al. (2006) and Ross (1999) for a review of both approaches), which
again nds empirical support (Mehlum et al. (2006) and Boschini et al. (2003)). Fi-
nally, there are studies addressing the reverse e¤ect the impact of the resource curse
on institutions. The e¤ect is found to be negative and is attributed to reduced govern-
ment accountability, a better ability to repressed opposition and increasing corruption
and rent-seeking (see Ross (2001), Sala-i-Martin and Subramanian (2003) and Collier
and Hoe er (2005)). We propose an alternative link, suggesting that the abundant
resources undermine the autocrats incentives to relinquish power and hence, hamper
capital accumulation.
Our model is consistent with a range of empirical ndings. It predicts that au-
Autocracy, Devolution and Growth 6
tocratic economies would be characterized by either relatively high or relatively low
growth rates, while the democratic regimes fall into an intermediate range, which is
in line with Almeida and Ferreira (2002). It also provides a new insight into the rela-
tionship between income and democratization. While many cross-country studies nd
that richer countries devolve earlier, the more recent panel data analysis documents
no e¤ect of income on democracy (e.g. Acemoglu et. al. (2005)) or on the hazard rate
out of autocracy (Persson and Tabellini (2006)) once the regressions include country
xed e¤ects. The model suggests an explanation for this result, showing that there
is no simple relation between countrys income and democratization. Indeed, income
in the model can be decomposed into the product of total factor productivity and
some function of capital. These di¤erent components of income are predicted to have
di¤erent e¤ect on timing of devolution: Higher initial capital leads to earlier devolu-
tion, while higher total factor productivity delays devolution. Hence, if this e¤ect of
decomposition of income is not accounted for (which is the case in empirical studies),
one would expect to nd no clear relation between income and democratization. Fur-
thermore, the model also suggests a new explanation to why many empirical studies
(e.g. Barro (1999), Alvarez et al. (2000), Boix and Stokes (2003)) nd that growth
causes democratization. In our model, the devolution of power is preceded by a more
or less extensive period of growth. Thus, if the data were to be generated according
to our model, we would nd that growth Granger causes democratization. Neverthe-
less, this conclusion is misleading: in the model, institutions of limited government
and growth are determined simultaneously and endogenously. Indeed, in the absence
of a possibility to relinquish power, growth would slow down if not completely stop.
Similarly, in the absence of growth opportunities, the institutions of limited govern-
ment would never be introduced. Therefore, by means of the model, we illustrate
why the observed time pattern between growth and democratization does not reect a
causal relation. This is in line with the more recent literature (Acemoglu et al.(2004),
Przeworski (2004)) that emphasizes the endogenous determination of institutions and
growth.
The paper proceeds as follows. Section 2 describes the setup of the model and
section 3 presents the models solution. Section 4 addresses the assumptions of the
model. The predictions of the model and its comparative statics are discussed in
section 5. Section 6 discusses relevant case studies. Finally, Section 7 concludes and
suggests some directions for further research.
2. The Model 7
2 The Model
An innite horizon economy ruled by an autocrat is populated by identical atomistic
citizens of mass one. At date t = 0, due to an exogenous shock, each citizen is being
exposed to a technology allowing for growth based on capital accumulation. In each
period t, output is produced from capital according to the Cobb-Douglas production
function
yt = A (Kt) ;
were Kt denotes the capital stock and yt denotes the output at time t. We assume
that capital depreciates completely in each period; that is, the depreciation rate is
1.
Each period, a citizen can stay in the market or leave the market and divert the
capital to an alternative activity. If some citizens stays, the Ruler has the power to
tax the citizens through an economy-wide consumption tax t. Taxation is costly for
the Ruler. We assume the cost of tax collection ( tct) is proportional to the tax base:
( tct) = ( t)ct;
where 0(:) > 0; "(:) > 0, and (0) = 0(0) = 0. The Ruler also has the option of
expropriating the entire production process (we assume technological indivisibility of
the capital stock) from each citizen. In the latter case, the Ruler can still employ the
Cobb-Douglas production function but, unlike the citizens, she is unable to accumulate
capital. Thus, due to complete depreciation, she can only use the expropriated capital
stock in one period.6 In case of expropriation, each citizen receives zero payo¤ from
the moment of expropriation and onwards.7
If a citizen leaves, she receives a payo¤ with the net present value of L, which is
non-taxable and non-expropriable. The decision to leave is irreversible: if diverted,
capital cannot be returned to the market sector. We assume that
L <1Xi=0
i ln(1 )AK
0
(1 + A); A1
6The introduction of a cost of expropriation does not a¤ect the qualitative predictions of the modelas long as the assumption of indivisibility of capital holds.
7The taxation scheme consisting of two instruments (consumption tax and complete expropria-tion of capital) was chosen to keep the analysis tractable. As will be shown below, in our setting,consumption tax does not inuence the capital accumulation path. This, together with the cost oftaxation, allows us to avoid the time inconsistency problem, standard in innite-horizon taxationmodels. We believe that the argument of the model will not be destroyed by replacing the proposedtax scheme by one distortive (capital) tax and classifying expropriation as unappropriately high levelof taxes. However, we are not able to solve for the resulting equilibrium.
Autocracy, Devolution and Growth 8
where A solves 1 0( i)(1 + i) + ( i) = 0: As we shall see, assumption A1 impliesthat the alternative activity is less e¢ cient than the market activity. That is, the
citizens payo¤ of leaving is less than the payo¤ she would get by staying in the
market, were the Ruler able to commit to never expropriate from her. We also assume
that the decision to leave is irreversible: if diverted, capital cannot be returned to the
market sector.
The Ruler can also choose to devolve her power. We assume that this decision is
equivalent to completely renouncing the right to expropriate and limiting the Rulers
power to set taxes; the Ruler cannot "ne tune" the devolution decision to keep all
current economic privileges.8 More precisely, we assume that after devolution of power,
the Ruler cannot set a consumption tax above some upper bound D; where
D < A: A2
We also assume that the devolution decision is irreversible, i.e., if the Ruler devolves
she cannot return. Moreover, we assume that in this case no other dictatorial ruler
can take over.
While being in power, the Ruler receives private benets of control of b units
each period. In the later point in the paper I will interpret these benets as natural
resource rents. After devolution of power, these benets are no longer available to
the Ruler. Note that we do not argue that the devolution deprives the Ruler of all
benets. Instead, b reects those benets of control that are lost upon devolution.
We also assume that if the economy reaches full development, the industrial sector
is su¢ ciently more productive than the natural resource sector, so the steady-state
post-devolution payo¤ of the Ruler exceeds the value of the ow of private benets.
The citizens instantaneous utility is logarithmic, so she maximizes the ow of her
future utilities
Vt =
1Xj=1
jv(ct+j) =
1Xj=1
j ln ct+j;
where ct is consumption and is the discount factor. The Rulers utility is linear and
her payo¤ function is denoted
Ut =
1Xj=1
jdt+j;
where dt+j is the payo¤ received by the Ruler in period t+ j.
The timing of the game is as follows: The citizens and the Ruler meet at discrete
time periods t = 0; 1; :::;1: Each period has three stages. If there was no devolutionof power in the past, at stage 1 of period t, the Ruler decides whether to devolve (D)
8In Section 4, we discuss the implications of relaxing this assumption.
3. Analysis of the Game 9
or abstain from devolution (ND). At stage 2, each citizen decides whether to stay in
the market sector (S) or leave (L). At stage 3 of period t; in case some citizens stay in
the market sector, the Ruler chooses whether or not to expropriate (E) the capital of
some citizens. Expropriated capital is used for production only in period t and then it
depreciates completely, as the Ruler cannot invest. If no expropriation (NE) occurs,
production, consumption and investment take place and taxes are paid. Then, the
game proceeds to the period t+ 1.
As in any other multi-stage game, the history of the game is a collection of all
actions played up to stage t. A (behavior) pure strategy of the ruler/citizen is a
function (ht)=(ht) prescribing an action in each of the rulers/citizenscontrolled
game tree nodes for a given history of the game ht. For example, conditional on the
history ht; at the rst stage of each period, a pure strategy of the ruler determines
a choice between Dt and NDt. At the second stage, the strategy of each citizen
determines St or Lt taking into account all the past actions including those just played
at stage 1 of period t. At the third stage, the rulers strategy denes a choice of
whether and whom to expropriate, again, conditional on all actions played up to stage
3 of period t and, if some capital is not expropriated, t; the non-expropriated citizens
strategy prescribes the investment choice for the next period.
In what follows we seek a symmetric (as regards the citizens) pure-strategy subgame-
perfect Nash equilibrium of this game. In addition, we assume that the equilibrium
actions of the player are constant on the histories that only di¤er in the behavior of
the sets of agents of measure zero (like in Gul, Sonnenschein and Wilson (1986)).9
This implies that the strategies of Ruler and each citizen cannot be conditioned on
the actions of any single citizen, or in other words, that unilateral deviation of a single
citizen does not a¤ect choices of the Ruler and the other citizens. With this assump-
tion, each citizen takes taxes and the timing of devolution as given. One can also show
that in the symmetric equilibrium the Ruler would not have incentive to behave di¤er-
ently with respect to di¤erent groups of citizens. These consideration allow us to take
a shortcut and discuss this game as a game between a Ruler and the representative
tax-taking Citizen.
3 Analysis of the Game
In this section we solve for symmetric subgame-perfect equilibria of the game. We
start by analyzing the development path of this economy in the absence of Rulers
9This assumption may require some technical restrictions, e.g. to insure that the set of citizenstaking each given action is measurable. We assume them to hold.
Autocracy, Devolution and Growth 10
D
ND
S
L
E
NER RC
t
D
ND
S
L
E
NER RC
t+1
Figure 1. Timing of the game
intervention, that is, without taxation, expropriation or devolution. We continue by
describing the optimal taxation choice of the Ruler, were she not able to expropriate,
and the choices of the Ruler and the Citizen along the non-expropriation path. Then
we proceed to show that for su¢ ciently high capital level the Ruler always prefers
expropriation to any other continuation strategy which allows us to treat this game
as a nite-horizon one and solve it backwards. Finally we discuss equilibria that arise
for di¤erent sets of parameter values.
3.1 The benchmark case: no government intervention
Let us start by describing the evolution of this economy in the absence of any govern-
mental intervention. As staying in the market sector is more productive than leaving
it, the e¢ cient outcome is for production to occur in the market sector and for the
citizen to maximize her utility
max
1Xj=1
j ln ct+j;
subject to the dynamic budget constraint
Kt+1 = A (Kt) ct:
3. Analysis of the Game 11
This problem is identical to the standard Ramsey growth model and the solution is10
ct = (1 )AKt ; (1)
Kt+1 = AKt :
The dynamic equilibrium of this economy is characterized by the capital and the output
monotonically converging to the steady state values K and y, respectively:11
K = (A)1
1 ;
y = A1
1 ()
1 :
Note that along the transition path, both capital and output are approaching the
steady state at a decreasing rate. The growth rate of capital
k(t) Kt+1 Kt
Kt
= AK1t 1;
and the growth rate of output
y(t) yt+1 ytyt
= [ k(t) + 1] 1;
are both decreasing in t:
3.2 Subgame-perfect Nash equilibrium
We turn to the analysis of the game between the Ruler and the Citizen and characterize
the pure-strategy subgame-perfect Nash equilibria of this game.
We start by examining the Rulers taxation incentives.
Lemma 1 If the Ruler were not able to expropriate and the Citizen were not able toleave the market, the Ruler would choose the same tax rate A each period, such that
1 0(A)(1 + A) + (A) = 0:
Proof. First, note that in our setting, consumption taxation does not inuence thecapital accumulation path (and thus the output path). Indeed, assume that the Citizen
is facing a ow of taxes f tg ; t = 1; :::;1: As was discussed above, in the equilibriumin consideration each citizen takes taxes as given - indeed, her unilateral deviation has
no impact on the choices of the Ruler and thus her investment decision could not a¤ect
taxes. Citizens problem then becomes
max1Xt=0
t ln ct
s.t. Kt+1 = AKt (1 + t)ct;
10It can be conrmed by the usual guess-and-verify method.11We concentrate on studying the increasing branch of the saddle path.
Autocracy, Devolution and Growth 12
and the solution is
ct =(1 )AK
t
(1 + t);
Kt+1 = AKt : (2)
Thus, it is only the consumption prole fctg that changes as compared to the bench-mark case (1), while capital accumulation is una¤ected by t.
As a result, at each point in time t, the Ruler solves the problem
maxf ig
1Xi=t
it ( ici ( ici)) =
maxf ig
1Xi=t
it ( i ( i))(1 )AK
i
(1 + i)
:
In the absence of the Citizens diversion opportunity, the solution of this maximization
problem does not depend on past taxes. In other words, the Ruler can decide on each
periods tax separately. The respective rst-order condition can be rewritten as
1 0( i)(1 + i) + ( i) = 0:
As > 0, the right-hand side of this equality is decreasing in i. Moreover, it is
positive at i = 0 and negative at i !1: Thus, there is a unique i A satisfyingrst-order conditions and, as second-order conditions are satised, it indeed determines
the maximum point.
Lemma (1) explains why we imposed assumption A1. The assumption ensures
that, if the Citizen leaves the market, she receives less than she would under Rulers
full discretion to tax (but not to expropriate). The Ruler in this case does not receive
any payo¤ in addition to the private benet of control, hence, overall e¢ ciency falls.
In turn, our assumption on post-devolutionary taxation A2 simply reects the idea
that devolution imposes some restrictions on the Ruler, so that she is no longer able
to choose her preferred tax rate.
We proceed to characterize the equilibria.
Lemma 2 In any SPNE in all subgames following the devolution of power, the Rulersets the highest possible tax rate D in each period and the Citizen stays in the market.
Proof. As shown above, a consumption tax does not inuence the capital accumu-lation path. Indeed, the citizens ignore the e¤ect of their decisions to stay or leave
on the Rulers choice of the tax rates and the timing of devolution, and take them as
given. Hence, in each period after the devolution of power, the Ruler and the Citizen
3. Analysis of the Game 13
"share a pie" of a predetermined size, unless the Citizen diverts the capital to the non-
market sector. Let us describe the symmetric subgame-perfect Nash equilibrium of
this subgame. First, note that given the restrictions on the post-devolution maximum
tax rate, the Citizen chooses to stay in the market for any tax schedule.. Indeed, after
devolution, the Ruler cannot not set the tax rate above D: So, at any point in time t,
the Citizens payo¤ from staying in the market is at least as high as her payo¤ when
the Ruler taxes at the rate D:
V S ( t; t+1; t+2; :::) 1Xi=t
it ln(1 )AK
i
(1 + D);
while her payo¤ from leaving the market is
V L = L:
Note that the Citizens payo¤ from leaving the market is below the post-devolutionary
payo¤. Indeed, as A > D and the capital stock increases over time,
1Xi=0
i ln(1 )AK
i
(1 + A)
1Xi=0
i ln(1 )AK
i
(1 + D)
1Xi=t
it ln(1 )AK
i
(1 + D): (3)
From our assumption (A1) and inequality (3), it follows that once devolution occurs,
it is never optimal for the Citizen to leave the market
L <1Xi=t
it ln(1 )AK
i
(1 + D):12
That is, after devolution of power, the Citizen always prefers staying in the market to
leaving for the alternative sector:
V S ( t; t+1; t+2; :::) > VL:
Thus, in any SPNE, the Ruler is free to choose the tax schedule that brings her
the maximal payo¤ given the post-devolutionary tax restrictions. As D < A, the
Rulers payo¤ increases with the increase in each periods tax. So in every period t,
she sets the tax rate t equal to the maximum post-devolutionary tax D.
Before proceeding we need to establish one important property of any (symmetric)
subgame-perfect equilibrium of this game. No SPNE can have expropriation on the
game path. Indeed, expropriation cannot occur in equilibrium, as the Citizen would
gain by deviating to the shadow sector just before the Ruler expropriates. It implies
12Here we assume that the capital level at which the Ruler chooses to devolve is at least as high asK0. We will see below that this assumption will always hold along the game path of the SPNE.
Autocracy, Devolution and Growth 14
that along the game path of any subgame-perfect Nash equilibrium each citizen, being
a tax-taker, invests a xed share of current output and this choice is independent of
actual tax rates (see Lemma 1). That is, the capital accumulation path always follows
rule (2). Therefore, in seeking the game path of SPNE we can limit ourselves to ana-
lyzing capital accumulation trajectories that correspond to this rule. This observation
together with Lemma (2) yields a useful corollary.
Corollary 1 Along the non-expropriation (sub)game path, the Citizen never leavesthe market. The Ruler chooses A while in power and D after devolution.
Proof. If there is no expropriation along the game path, the maximum tax the Rulerwould set is A: Thus, the Citizens payo¤ from staying will never be below
V =1X
i=t
it ln(1 )AK
i
(1 + A):
But, as capital accumulates, the payo¤ from leaving is always below the payo¤ from
staying:
L <1Xi=0
i ln(1 )AK
i
(1 + A)
1Xi=t
it ln(1 )AK
i
(1 + A)= V V S ( t; t+1; t+2; :::) :
As a result, along the non-expropriation game path, the Ruler can always choose his
preferred tax levels, that is A while in power and D after devolution.
Denote the share of output the Ruler receives while being in power by
"A =A (A)(1 + A)
(1 );
and the one she gets after devolution by
"D =D (D)(1 + D)
(1 ):
Note that as the tax in autocracy is higher than after devolution, A > D; the same
is true about the Rulers output shares:
"A > "D:
We are now ready to study how the Rulers incentive to expropriate evolves over
time.
Lemma 3 If "A(1) < 1; then in any SPNE along all the paths where the Citizen invests
a xed share of current output, there exists a nite time period T such that the
Ruler prefers expropriation at stage 3 of period T over any other continuation strategy.
3. Analysis of the Game 15
Proof. We start with some introductory observations. There are three types of "exits"in this game. First, the Ruler can expropriate the capital from the Citizen. If this
occurs, the Citizen is left with zero capital and cannot restart production, so that
the continuation of the game is fully predetermined. Second, the Citizen can divert
the capital to the shadow sector, which makes the continuation game independent
of the players actions. Finally, the Ruler can devolve power. Since this decision
is irreversible, and after devolution the continuation game has a unique subgame-
perfect Nash Equilibrium, the value of the remaining game is also clear at the time of
devolution.
Observe that after the capital has reached a su¢ ciently high level, no SPNE can
have exits of the rst two types along the game path. Indeed, in case of expropriation,
the Citizen gets zero, while if she diverts the capital at the preceding stage, she is
guaranteed a positive payo¤. Similarly, if the Citizen diverts the capital, the Ruler is
left with the private benets of control only. As we assume that the steady-state post-
devolution payo¤ of the Ruler is higher than the value of the ow of private benets,
by continuity, the same holds if devolution occurs su¢ ciently close to the steady state.
Thus, the Ruler prefers devolving power at the preceding stage rather than clinging
to power and only receiving the benets of control.
This implies that in the subgame starting at stage 3 of some period t the Ruler is
choosing between immediate expropriation, set of continuation paths where the game
"ends" by the devolution of power at some future period t + et and a path where thegame "continues" forever. The assumption of the Lemma ensures that in the steady
state the net present value of tax returns is not high enough to prevent the Ruler from
expropriating. By continuity, the same result holds around the steady state.
This Lemma states that if the agents are not very patient, the non-market activity
is not very ine¢ cient, and/or the share of capital in production is high, expropriation
always takes place if the economy is su¢ ciently close to the steady state. Note that if
the Rulers value of expropriating at T is higher than the value of any alternative con-
tinuation strategy at time T , the same is true for any t > T , as output monotonically
converges to the steady-state value y.
Corollary 2 If "A1 < 1; and T is the time period found in Lemma 3, then any
SPNE is characterized by the Ruler choosing to expropriate at stage 3 of every period
T; T + 1; T + 2; :::along all the paths where the Citizen invests a xed share ; and
the Citizen leaving the market to avoid expropriation.
Throughout the paper, we assume that the conditions of Lemma 3 hold; that is,
the parameters of the model are such that in the steady state, the Ruler always prefers
Autocracy, Devolution and Growth 16
expropriation over any other strategy. Corollary 2 thus allows us to treat period T as
a "nal" period and solve the game backwards from there.
Period T As shown in Corollary 2, at stage 3 of period T , the Ruler expropriates.
At stage 2 of period T , the Citizen leaves the market sector to get a positive payo¤,
as compared to the zero payo¤ in case of staying and being expropriated at the next
stage. As a result, at stage 1 of period T , the Ruler has to choose between devolution,
in which case she receives a payo¤
U (DT ) D1Xi=0
icT+i = "D
1Xi=0
iyT+i;
and no devolution, which gives her a ow of future benets of control
U (NDT ) 1Xi=0
ib =b
1 ;
as the Citizen leaves the market at the next stage.
Period T-1 At stage 3 of period T 1, the Ruler compares the option of expropri-ating and getting
U (ET1) = yT1 +b
1 to the option of non-expropriating, taxing and proceeding to stage 1 of period T (where
she either devolves or stays and receives a ow of benets of control only).
If already in period T , the devolution payo¤ is lower than the ow of the benets
of control
U (DT ) < U (NDT ) , (4)
the Ruler has to decide whether to expropriate at stage 3 of period T 1 and receiveU (ET1 ), or proceed to stage 1 of period T to get the ow of the benets of control
and receive
U (NET1; A; NDT ) = "AyT1 + b+ U (NDT ) = "AyT1 +b
1 :
Since
U (NET1; A; NDT ) = "AyT1 +b
1 < yT1 +b
1 = U (ET1 ) ;
the Ruler prefers to expropriate at T1 as well. Note that the devolution payo¤U(Dt)
is increasing over time, while the payo¤ of sustaining the autocratic regime with no
market production U (NDt ) is constant. Thus, if inequality (4) holds, that is, the
3. Analysis of the Game 17
devolution payo¤ is lower than the accumulated private benet in period T , the same
is true for all periods t < T . As a result, the backward induction argument repeats
itself until period t = 0, where at stage 3 the Ruler again chooses to expropriate, at
stage 2 the Citizen leaves the market and at stage 1, the Ruler stays to enjoy the
ow of private benets of control. The Rulers incentive to keep the private benets
of control is so strong that she never wants to devolve and lose them. Therefore, the
Ruler cannot commit not to expropriate and, as a result, no investment ever takes
place.
Proposition 1 An economy with su¢ ciently high private benets of control b is stuckin an "underdevelopment trap": the autocratic ruler never relinquishes power and the
e¢ cient production technologies never get implemented.
Assume now that the private benets of control are not too high, so that devolution
is chosen at stage 1 of period T ,
U (DT ) > U (NDT ) :
Then, the Ruler compares the payo¤ of expropriation U (ET1 ) to the devolution
payo¤ in period T
U (NET1; A; DT ) = "AyT1 + b+ U (DT )
= "AyT1 + b+
""D
1Xi=0
iyi+T
#:
She prefers expropriation i¤
U (NET1; A; DT ) < U (ET ) : (5)
Suppose that condition (5) is met so that the ruler chooses to expropriate. Similarly
to above, at stage 2 of period T 1, the citizen prefers to leave the market. At stage1, the ruler once more either devolves to prevent the citizen from leaving, and gets the
payo¤
U (DT1) = "D
1Xi=0
iyi+T1;
or does not devolve and receives the benet of control
U (NDT1 ) 1Xi=0
ib =b
1 :
As we have already discussed what happens in the case when the latter exceeds the
former, assume now that
U (DT1) > U (NDT1) :
Autocracy, Devolution and Growth 18
Period T-2 At stage 3, the Ruler again faces the choice of expropriating the Citizen
and getting
U (ET2) = yT2 +b
1 ;
vs. proceeding to stage 1 of period T 1 where she devolves, which yields
U (NET2; A; DT1) = "AyT2 + b+ U (DT1 )
= "AyT2 + b+ "D
1Xi=0
i+1yi+T1:
As above, the Ruler expropriates i¤
U (NET2; A; DT1) < U (ET2) : (6)
The backward induction proceeds to stage 2 of period T 2, where the argumentrepeats.
Before we proceed, we need to establish a useful result.
Lemma 4 For a given set of parameters, the di¤erence U (NEt; A; Dt+1)U (Et) issingle-peaked with a peak at some nite et.Proof. See the Appendix.As shown above, the Rulers devolution decision is determined by the di¤erence
between her payo¤ from expropriation in period t and her payo¤ from devolution in
period t+1. This Lemma establishes that there is a unique period of time et where therespective di¤erence reaches its maximum and that it monotonously declines for t > et.Continuing to solve the model backwards, there are two possible scenarios: (i)
either in period et, the expropriation payo¤ falls below the non-expropriation payo¤U (Eet) < U NEet; A; Det+1
or (ii) in each period t 0, expropriation is preferred over non-expropriation. Denotethe set of parameters supporting case (i) by DD, (for the reasons that will be clear
below) and the set of parameters supporting case (ii) by 2. It is easy to see that
neither of these cases is degenerate, that is, that both sets PD and 2 are non-empty.
Indeed, assume that b = 0, and consider the di¤erence between the payo¤s from
expropriation and non-expropriation (followed by devolution) in the initial period t =
0. As output is increasing over time,
U (NE0; A; D1) U (E0) = "D
1Xt=1
tyt (1 "A)y0 > "D1Xt=1
ty1 (1 "A)y0
= y0
"D
1 y1y0 (1 "A)
: (7)
3. Analysis of the Game 19
For given values of ("A; "D; ; A), if the initial capital is su¢ ciently low, the growth
rate in the rst periody1y0=AK
(1)0
is su¢ ciently high for expression (7) to be positive. Continuity ensures that the same
holds for small positive b. By Lemma 4
UNEet; A; Det+1 U (Eet) U (NE0; A; D1) U (E0) > 0;
which proves the nonemptiness of the set PD.
Alternatively, consider the peak period et. As output increases towards the steady-state value y,
UNEet; A; Det+1 U (Eet) = "D
1Xi=1
iyet+i (1 "A)yet b
1
< "D
1Xi=1
iy (1 "A)yet b
1 : (8)
Note that due to the separability of the payo¤ function, et is independent of b. Given("A; "D; ; A;K0), one can choose a su¢ ciently high b so that
0 < "D
1Xi=1
iy b
1 < (1 "A)yet:
Hence, expression (8) is negative, i.e., the considered parameters belong to the set 2.
We proceed by analyzing these two cases separately.
Case iBy construction, the payo¤ of expropriation in et is less than the payo¤ of devolution
in et + 1. Moreover, we know that the Ruler prefers expropriation to devolution at
the steady state, i.e., when t ! 1. By Lemma 4, there exist bt et such that theRuler prefers to expropriate at stage 3 of any period t > bt, but not to expropriate atstage 3 of period bt. It implies that the Ruler devolves at stage 1 of period bt+ 1. Thereason is simple: if the Ruler were to stay in power at stage 1 of period bt+ 1, by non-expropriating at stage stage 3 of period bt, she would receive a ow of private benetsof control as well as tax revenue in period bt. Thus, her payo¤ from not expropriating inbt falls short of the expropriation payo¤ as the latter includes both the private benetsand the entire output in period bt. That is,
UNEbt; A; Dbt+1 > U (Ebt ) : (9)
Autocracy, Devolution and Growth 20
By backward induction, the Citizen stays in the market at stage 2 of period bt.Indeed, no expropriation takes place at stage 3 and the devolution occurs at stage 1
of period bt + 1. Hence, there is no expropriation along this (sub)game path and, byCorollary 1, the Ruler sets bt = A, and the Citizen chooses to stay. At stage 1 of
period bt, the Ruler compares the option of devolving to the option of proceeding tothe next stage. Devolution yields the Rulers payo¤
U (Dbt) = "D1Xi=0
iAKi+bt: (10)
Non-devolution followed by taxation entails a positive tax revenue in this period and
the devolution payo¤ from the next period on
UNDbt; NEbt; A; Dbt+1 = "AAKbt + b+ "D
1Xi=1
iAKi+bt: (11)
As taxes under autocracy are higher than after devolution, "A > "D and b 0, the
Ruler does not devolve at stage 1 of period bt.Before proceeding backwards to period bt 1, we need to establish an intermediate
result. Consider the Rulers net payo¤
Lemma 5 As time passes, the Rulers payo¤ from expropriation net of private benetsof control becomes more attractive relative to the payo¤ from devolution net of private
benets of control.
Proof. See the Appendix.Intuitively, the further away is the economy from the steady state, the longer is the
growth horizon and the more appealing it is to get a share of future increasing prots
(through devolution of power), as compared to grabbing the entire pie today.
Now, we are ready to discuss the choice of Ruler in the period bt 1.Lemma 6 At stage 3 of period bt 1, the Ruler prefers non-expropriation over expro-priation.
Proof. See the Appendix.By denition, at stage 3 of period bt, the Ruler prefers non-expropriation, followed
by devolution, to expropriation. The Rulers choice between non-expropriation and
expropriation is determined by two factors: the growth rate of output and the private
benets of control that are lost upon devolution. Now turn to period bt 1. By
choosing not to expropriate in period bt 1, the Ruler retains the benets of controlfor period bt, as she does not devolve in that period. Therefore, as compared to period
3. Analysis of the Game 21
bt, expropriation at bt 1 is associated with a smaller gain in the private benets ofcontrol. In addition, the growth rate of output in period bt 1 is higher than at bt,thereby providing the Ruler with additional incentive to refrain from expropriation.
Hence, the Ruler chooses not to expropriate at stage 3 of period bt 1,UNEbt1; A; NDbt; NEbt; A; Dbt+1 > U Ebt1 :
As above, at stage 2 of period bt 1, the Citizen decides to stay. At stage 1 ofperiod bt 1 the Ruler again chooses whether to devolve. If she does not devolve, herpayo¤ is
UNDbt1; NEbt1; A; NDbt; NEbt; A; Dbt+1 = "AAK
bt1 + b+ "AAK
bt + b
+
1Xi=1
i"DAKi+bt:
An immediate devolution yields the same ow of payments from period bt+1 onwards,but lower payo¤s in periods bt 1 and bt
U (Dbt) = "DAKbt1 + "DAKbt +1X
i=1
i"DAKi+bt:
Therefore, the Ruler chooses not to devolve at stage 1 of period bt 1. At stage 3 ofperiod bt 2, we repeat the argument of Lemma 6 to conclude that the Ruler againprefers not to expropriate.
We continue solving the model backwards along the same lines until we reach
period t = 0. In all these steps, the optimal strategy for the Ruler is to tax the citizen
without devolution or expropriation. The optimal strategy for the Citizen is to stay in
the market sector. The tax rate chosen by the Ruler is set to t = A in each period
t = 1; 2; :::bt 1:Thus, we can conclude that there is a unique symmetric pure strategy SPNE where
along the game path the Ruler taxes the Citizen while retaining autocratic power and
not expropriating up to period bt; and devolves in period bt; that is(ht) =
0@ (NDt; NEt) ; t = 0; :::;bt; Dbt+1; t =
A; t = 0; :::;bt;D; i = bt+ 1; :::;1
1A :The citizen always stays in the market along the game path:
(ht) = (St; t = 0; ::1) :
That is, for each set of parameters in DD the Ruler chooses to delay devolution. We
summarize our ndings in the following proposition:
Autocracy, Devolution and Growth 22
Proposition 2 There exists a non-empty set of model parameters DD, such thatthe Ruler prefers to devolve rather than expropriate in at least one period et. If theparameters belong to the set DD, then along the game path of the unique symmetric
pure strategy SPNE, the Ruler delays devolution until period bt, where bt is the latestperiod in which the Ruler prefers devolution over expropriation.
Case iiIn this case, in each period t 0 expropriation is preferred over non-expropriation
followed by devolution in the next period. The resulting equilibrium depends on the
relation between the Rulers payo¤ to devolution in the initial period and the value of
the future ow of private benets of control.
More precisely, suppose that the parameter values are such that in period t = 0,
the Rulerss valuation of immediate devolution exceeds the valuation of the ow of
control benets:
U (D0) >b
1 : (12)
As the payo¤ to devolution increases over time, the same holds for every subsequent
period t > 0 :
U (Dt) >b
1 :
Thus, when we solve the game backwards, at stage 3 of every period the Ruler prefers
to expropriate, at stage 2 the Citizen leaves the market and at stage 1 the Ruler
chooses to devolve. This also holds for the very rst period t = 0, which means that
the devolution occurs immediately, before any growth in this economy takes place.
Here, the Ruler values growth after devolution more than stagnation in an autocratic
regime. To achieve any growth, she must relinquish her power in the very rst period,
otherwise the citizen immediately switches to a non-market activity.
In this case, in the unique symmetric pure strategy SPNE, the ruler devolves in
the initial period t = 0; that is
(ht) =
D0;
t = D; t = 0; ::1
:
The citizen again never leaves the market sector along the game path:
(ht) = (St; t = 0; ::1) :
Conditions (10) and (12) suggest that such an equilibrium can take place in economies
where e.g. the initial capital is relatively high, the private benets of control are mod-
erate, or the Rulers power to tax after devolution is relatively large. By construction,
the set of parameters supporting this equilibrium is a subset of 2. Denote this set by
3. Analysis of the Game 23
ID, where ID stays for immediate devolution. This set is non-empty. For example,
this outcome can be observed in economies that start up very close to the steady state.
As shown above, the Ruler always prefers to expropriate; the e¢ ciency assumption en-
sures that the ow of private benets of control falls short of the devolution payo¤ in
the steady state which, by continuity, also holds in a neighborhood of the steady state.
Proposition 3 There exists a non-empty set of model parameters ID, such that theRulers payo¤ from expropriation is always higher than the payo¤ from the devolution
in the next period, but the payo¤ from devolution in t = 0 exceeds the value of the
ow of private benets of control. If the parameters belong to the set ID, devolution
occurs in the very rst period and the economy fully realizes its growth potential.
Finally, consider the situation where, in period t = 0, the Rulers payo¤ of imme-
diate devolution is lower than the net present value of the ow of control benets:
U (D0) <b
1 : (13)
Here, when we solve the game backward, the Ruler still expropriates at stage 3 of
each period t. This holds in period T by Lemma 3. It also holds in all earlier periods
irrespective of the Rulers choice at stage 1 of period t + 1: Therefore, the Citizen
again leaves at stage 2: But now the Ruler prefers not to devolve at stage 1 of early
periods, as devolution is not su¢ ciently attractive. In particular, there is no devolution
in the initial period t = 0where inequality (13) holds. Therefore, such an economy
also becomes locked in an underdevelopment trap: at the beginning of growth, the
post-devolutionary future does not look su¢ ciently tempting to the Ruler. Thus, she
prefers to retain all her political power to receive the private benets of control, even
though the Citizen immediately leaves the market to avoid expropriation. As a result,
growth in this economy never occurs.
Formally, in the unique symmetric pure strategy SPNE along the game path, the
ruler does not devolve in period t = 0:
(ht) = (ND0) ;
and the citizen leaves the market sector in the very initial period:
(ht) = (L0) :
Similarly to the above, conditions (10) and (12) suggest that this equilibrium outcome
is observed in economies with low initial capital, high benets of control and relatively
limited Rulers power to tax after devolution. Denote the respective set of parameters
by U 2.
Autocracy, Devolution and Growth 24
Proposition 4 There exists a non-empty set of model parameters U such that theRulers payo¤ from expropriation is always higher than the payo¤ from the devolution
in the next period, and the value of the ow of private benets of control exceeds the
payo¤ from devolution at t = 0. If the parameters belong to the set U , the economy
is caught in an underdevelopment trap: no growth or devolution of power ever occurs.
4 Discussion
In this section, we discuss the key assumptions of our model and their implication for
the results. In the model, we abstract from open conict. That is, the only threat
that the Citizen can make to the Ruler is that of exit. Introducing the possibility
of conict into the model would not change the nature of its predictions, but might
bring some additional insights. Assume that the Citizen can struggle with the Ruler
in order to force her to relinquish the power. Then, as growth declines and the Ruler
is prepared to give up power in the near future, we may expect both the Ruler and
the Citizen to ght less intensively. Thus, unlike in a stationary setting where the
intensity of struggle would typically be constant, we may nd that as growth slows
down, uproars become less violent. This extension could be helpful in relating the
model to the evidence, which suggests that even peaceful devolutions of power are
normally preceded by some pressure on the ruler.
Another key assumption is that the ruler can commit to relinquish the power. In
other words, the institutional structure in the economy allows for the devolution of
power. The formation of these institutions is beyond the scope of our analysis. Clearly,
if such institutions are lacking, the Ruler cannot credibly commit not to expropriate.
Similarly, we assume that the citizens can, in turn, guarantee the ruler a "safe haven"
after she devolves. If such an institution is missing, the autocrat has no incentive to
devolve. Therefore, in the absence of either of these institutions, the model would
predict any economy to be locked in an underdevelopment trap.
What happens if we allow the Ruler to "ne tune" the devolution decision? That
is, suppose the Ruler may choose the post-devolutionary tax level (while keeping the
assumption that devolution is associated with the absence of expropriation). Un-
der this relaxed assumption, the Ruler does not impose any restrictions on the post-
devolutionary tax return. The devolution of power occurs in any period between t = 0
and the period when the payo¤ of expropriation is just below the payo¤ of eternal
taxation. Indeed, the Ruler can now make devolution as protable as taxation and is
thus indi¤erent concerning the timing of the devolution, as long as the Citizen does
not leave the market. For example, if the Citizen has a weak preference for devolution,
5. Predictions of the model 25
the Ruler is ready to devolve in the very initial period. That is, such "ne tuning"
prevents us from predicting the precise timing of devolution. However, as long as
the "ne tuning" implies post-devolutionary loss in private benets of control, the
ine¢ cient "underdevelopment trap" equilibrium outcome continues to exist.
Some of the less realistic predictions of the model are artifacts of simplication. For
example, along the equilibrium path, no expropriation occurs in the model, while we
do observe examples of the governments predative behavior in real life. Lack of expro-
priation in the model is due to the fact that we have assumed perfect information and
no uncertainty. If we instead assume that the Citizen is imperfectly informed, or there
are random shocks to the production function, we extend the set of SPNE by includ-
ing equilibria involving expropriation of capital along the equilibrium path. Similarly,
uncertainty can yield "revolutionary" equilibria, that is, equilibria with conict-driven
devolution of power.
In our pure-strategy SPNE, no autocracy can survive in the long run, while we do
observe non-collapsing autocracies in the real world (e.g. consider China). However,
this does not imply that the model contradicts the evidence. The growth prospects
might be su¢ ciently good, so that the devolution stage has not yet been reached.
Also, we have deliberately conned the attention to a set of parameters under which
expropriation is preferred in the steady state. Relaxing this assumption can produce
equilibria with an eternal growing autocracy.
Finally, the model predicts that the growth rate declines after the devolution of
power. Note that in our model, the devolution of power is, in fact, represented by an
improvement in property rights protection. Most empirical studies record the opposite
e¤ect better property right protection spurs growth. This e¤ect can easily be incor-
porated in the basic framework. For example, we might extend the model to allow
for more sectors. If some sectors do not have a diversion opportunity, these sectors
will start to accumulate only in the absence of the expropriation threat. That is, the
devolution of power will give rise to an additional wave of investment and growth, not
attainable under autocracy.
5 Predictions of the model
Now we are ready to discuss the models predictions and comparative statics, as well
as the importance of our modelling assumptions.
Proposition 5 In an economy that is growing under autocratic rule, a higher level ofinitial capital entails earlier devolution of power. In an economy locked in an under-
Autocracy, Devolution and Growth 26
development trap, an increase in initial capital may entail early devolution of power
and growth.
The formal proof can be found in the Appendix. Informally, consider two economies,
one starting with the initial capital K0 and another - with the capital that the rst
economy would reach in period t = 1,
K0
0 = K1 > K0:
The backward induction procedure described above implies that the choices made by
the agents in the rst economy in period t are identical to the choices made by the
agents in the second economy in period t 1. Thus, if in the former economy thedevolution of power occurs at date bt, in the latter economy it occurs at date bt 1.Intuitively, an increase in the initial capital, other things equal, implies that the econ-
omy starts closer to the steady state and experiences lower growth rates throughout
its development path. As a result, the future does not look that tempting for the Ruler
and her incentive to grab at each point in time increases. Thus, in order to persuade
the citizens to remain in the market, the Ruler needs to devolve power earlier.
Alternatively, assume that the former economy is in the "underdevelopment trap".
That is, in any period t > 0, the Ruler prefers expropriation at stage 3 of period t
to devolution at stage 1 of period t + 1 and her devolution payo¤ in period t = 0 is
less than the ow of private benets of control. Then, an increase in initial capital to
K00 = K1 does not inuence the relative attractiveness of expropriation, as compared
to the devolution in the next period. Indeed, as mentioned above, the decisions made
in the latter economy at time t replicate the decisions made in the former at time
t + 1, so the Ruler of the economy starting with K00 still prefers expropriation at any
t0 > 0. Therefore, the institutions are fully determined by the Rulers devolution
decision in period t = 0. Note that the Rulers devolution payo¤ is increasing in the
level of the initial capital, while the ow of private benets of control is constant. As
K00 = K1 > K0, the devolution of power in period t = 0 in the latter economy brings
the Ruler as much as the devolution in period t = 1 in the former economy, which is
higher than the payo¤ to devolution in the initial period in the former economy:
U (D0jK 00) = U (D1jK0) > U (D0jK0) :
Hence, it may be the case that the devolution payo¤ at K00 exceeds the ow of private
benets of control and thus, in the latter economy, the devolution of power occurs in
the very initial period t0 = 0.
The above discussion implies the following corollary.
5. Predictions of the model 27
Corollary 3 Economies with lower level of initial capital are more likely to be lockedin an underdevelopment trap.
An underdevelopment trap equilibrium outcome can only arise in an economy where
the set parameters (;A; ; D; A; b) 2 U , so that the Ruler never prefers devolutionover expropriation. As shown above, among these economies, higher initial capital
leads to early devolution and growth, while lower capital blocks the economic and
institutional development.
Now consider a technological change an increase in the total factor productivity
parameter A. Intuitively, a country with a higher total factor productivity has a higher
growth rate in each period and steady-state capital. So, at each point in time, this
countrys future growth potential weakens the incentives to expropriate. As a result,
we expect the devolution of power to be delayed. On the other hand, the value of
devolution relative to the value of expropriation increases with TFP (e.g. due to a
higher growth rate in each period). Thus, higher total factor productivity may improve
the chances for eventual devolution in economies in an underdevelopment trap.
Proposition 6 If an economy is in the underdevelopment trap, higher total factor pro-ductivity may create the devolution of power and growth. For two growing economies,
an economy with higher total factor productivity, other things equal, experiences later
devolution of power.
Proof. See the Appendix.Other things equal, higher total factor productivity translates into a higher growth
rate. Hence, we have an immediate Corollary:
Corollary 4 Among growing economies, autocracies are more likely to experiencehigher growth rates than are less autocratic regimes.
An increase in labor intensity or in discount factor has an ambiguous e¤ect.
Comparing the results of Propositions 5 and 6 we see that di¤erent components of
initial income have opposite e¤ects on the timing of the devolution of power - higher
initial capital causes earlier devolution while higher total factor productivity delays it.
This may explain why recent panel-data studies nd no e¤ect of income on democracy
(e.g. Acemoglu et. al (2005)) or on the hazard rate out of autocracy (Persson and
Tabellini (2006)) once they control for country xed e¤ects. Indeed, the model predicts
a non-linear e¤ect of the two components of income, to capture which one would need
to include an interaction term between income and an indicator for high TFP into
a democracy regression. This interaction term would have a negative predicted sign.
Autocracy, Devolution and Growth 28
On the other hand, a long-run e¤ect between the income and growth that arises in
the Acemoglu et. al (2005)s regression (and produces the well-known cross-country
correlation between income and democracy) reects di¤erent development paths for
di¤erent countries. In our model it would be captured by the set of the exogenous
(including institutional) factors, such as the existence of the devolution mechanism,
private benets of control etc.
Next, let us address the impact of private benets of control on the devolution of
power. The Ruler retains the benets of control only while being in power, whereas
these benets are no longer available to her after devolution. Thus, private benets do
not inuence the Rulers trade-o¤ between early and late expropriation. Instead, they
only have an impact on the Rulers incentive to devolve. If the benets are low, the
Ruler can delay devolution for a long time, because the Citizen realizes that the Ruler
will not cling to power the day she needs to commit not to expropriate to avoid losing
investment. On the contrary, if private benets of control are very high, the Ruler will
not be willing to ever give up power, no matter how much capital accumulation is lost.
Recognizing this, the Citizen never invests in the market sector and there is neither
devolution nor growth. Thus, an abundance of natural resources has a detrimental
e¤ect on growth. In the intermediate range, as the private benet of control increases,
the Rulers relative incentive to expropriate, as opposed to devolution, increases too.
As a result, in order to keep capital accumulation going, the Ruler needs to devolve
earlier since, at later stages, she always prefers to stay in power, and no commitment is
possible. Thus, in this range, an increase in private benets speeds up the devolution
of power. Therefore, the model predicts a non-linear e¤ect of the private benets
of control on the devolution of power. This prediction of the model parallels the
arguments of the resource curse literature that abundant natural resources may
hinder growth13 and capital accumulation.14 Moreover, it suggests that the resource
curse should also be non-linear: an increase in resource size does not have any impact
on growth, until the resource rents become su¢ ciently large to completely kill growth.15
We summarize our ndings in the following Proposition.
Proposition 7 If the Rulers private benets of control are su¢ ciently small, an in-crease in the private benets causes earlier devolution of power. Eventually, a further
increase in the private benets of control locks an economy in an underdevelopment
trap with neither growth nor devolution.
13E.g. see Sachs and Warner(1995) and Gylfason (2001)14See Gylfason and Zoega (2001)15A non-linear (negative) e¤ect of the resource curse on growth is found by e.g. Sala-i-Martin and
Subramanian (2003)
5. Predictions of the model 29
Proof. See the Appendix.Note that the maximum post-devolutionary tax D available to the Ruler and the
private benets of control are two sides of the same coin. That is, both of them reect
the Rulers loss associated with giving up power. Thus, the e¤ect of D should be
similar to that of the benet of control. Indeed, a higher post-devolutionary tax rate
makes the devolution option more attractive relative to the expropriation. Thus, the
Ruler can credibly postpone devolution without jeopardizing industrialization. On the
other hand, if the Rulers payo¤ after devolution is very low (or very uncertain), she
has no incentive to devolve and the economy is in the underdevelopment trap.16 For
such an economy, a su¢ cient increase in D will cause an eventual devolution of power.
Proposition 8 If the post-devolutionary tax rate D is su¢ ciently low, the economyis locked in the underdevelopment trap. An increase in the post-devolutionary tax rate
rst causes devolution to occur and then delays it.
Proof. See the Appendix.Similarly, a change in the cost of taxation (:) inducing an increase in A and
Rulers tax revenues received under autocracy has the same impact on the timing of
devolution as an increase in D. (Here we only consider an increase in A which does
not change the Rulers incentive to expropriate in the steady state, and the incentive
of the citizen to stay in the market as long as there is no expropriation, so that
the assumption of Lemma 3 and condition (7) continue to hold). This result has a
very simple explanation. When the Ruler decides whether or not to expropriate, she
weights the expropriation payo¤ against the payo¤ from non-expropriation today and
devolution next period. If she chooses to devolve, she retains the todays autocratic
tax revenue and receives the devolution payo¤ from tomorrow onwards. Therefore,
by expropriating at period t, she foregoes the tax revenue "Ayt. The higher is this
revenue, the weaker are her incentives to expropriate and the longer she can stay in
power without using the commitment device. On the other hand, if the economy is
locked in an underdevelopment trap, the tax revenues from capital accumulation under
autocratic regime are too low to persuade the Ruler to forgo the private benets of
control. In this case an increase in A may bring about an eventual devolution.
Proposition 9 If the autocratic tax rate A is su¢ ciently low, the economy is locked inthe underdevelopment trap. In this case a change in the cost of taxation (:) inducing
an increase in A causes eventual devolution of power. In a growing economy an
increase in A delays devolution..16E.g. consider an extreme case when the Ruler cannot be credibly guaranteed any post-
devolutionary payment from the citizen.
Autocracy, Devolution and Growth 30
Proof. See the Appendix.To illustrate our ndings, we consider numerical simulations with parameters A =
1; = 0:36; D = 0:35; A = 0:4 and = 0:7.17 In gure 2, we graph the set of
ΩID
ΩDD
ΩU
Figure 2. Development trajectories
equilibria in our economy as a function of the initial capital and the private benets
of control. We see that low values of initial capital and private benets of control
result in an equilibrium with delayed devolution of power. An increase in either ini-
tial capital or the benet of control brings the economy into the area of immediate
devolution of power. A further increase in the private benet of control leads to the
"underdevelopment trap" equilibria with neither growth nor devolution of power.
What are the predictions on the relationship between growth and the political
regime? In our model, depending on the parameter values, there are two possible
regimes: Either an economy is locked in the underdevelopment trap and neither growth
nor devolution occurs, or the economy sustains an autocratic regime at higher growth
rates and devolves as growth slows down. Thus, we see that the dictatorships are char-
acterized either by no growth or by high growth rates, while less autocratic regimes fall
in an intermediate range. The source of this cross-sectional variation may be di¤erent17The values of A and a are standard for numerical simulations of the Ramsey model. The value
of the time discount captures the fact that in our model we have complete depreciation over oneperiod.
6. Some cross-country and case studies 31
initial conditions, di¤erent stages of development (that is, the time of acquiring the
technology) or a di¤erence in technology per se. This prediction is consistent with the
nding of Almeida and Ferreira (2002), who show that the cross-country variability
of growth rates is higher among autocracies, and that autocracies are likely to be the
best and the worst performers in terms of growth.
Second, according to the model, devolution of power is often preceded by several
periods of growth. Therefore, it may look as if, in line with the results of Barro (1999)
and others, the model establishes a causal relationship from growth to democratiza-
tion, at least in the Granger-sense. But this conclusion is misleading: in the model,
institutions of limited government and growth are determined simultaneously and en-
dogenously. Indeed, if the institutions facilitating the devolution of power are missing
in an economy, so that the power cannot be credibly relinquished, growth will not
occur. On the other hand, in the absence of growth opportunities, the government
would never self-impose any checks and balances. Therefore, the observed time pattern
between growth and democratization does not necessarily reect a causal relation.
6 Some cross-country and case studies
In this section, while not aiming at systematic empirical analysis, we relate the pre-
dictions of the model to a few observed patterns. As a preliminary step, we try to
replicate the pattern of Figure 2, which links the initial capital and private benets of
control to the institutional transformation. For a set of countries, we plot the proxy
of the initial capital against the proxy of the private benets, grouping the countries
by the time of their rst institutional improvement after 1820. We approximate initial
capital by the logarithm of a countrys GDP per capita in 1820, taken from "Historical
Statistics for the World Economy" by Madisson (2006), where year 1820 is chosen as
the earliest year for which the GDP data is available for a decent number of countries.
Private benets of control are approximated by the countrys abundance of natural
resources, which is in turn proxied by the share of natural resources in total export
in 1970 taken from Sachs and Warner (1995). We dene institutional improvement as
the increase by a minimum of 3 units in the Constraints on Executive variable from
Polity IV data set. The result is presented in Figure 3.
Autocracy, Devolution and Growth 32
Figure3.
Institutionalimprovementismeasuredasanincreasebyaminimum
of3unitsintheConstraintsonexecutive(XCONST)variablefrom
PolityIVdataset.Itrangesfrom
1to7,withhighervaluescorrespondingtomoreconstrainedexecutive.CountriesthathadXCONST4
bytheinitialtimeofentryintothePolityIVdatabaseareclassiedashavingtheirrstinstitutionalimprovementatthemomentofentry.
InitialcapitalisproxiedbythelogofacountrysGDPpercapitain1820(Maddison(2006))
Privatebenetsofcontrolaremeasuredbythecountrysabundanceofnaturalresources(whichisproxiedbytheshareofnaturalresources
intotalexportin1970).
6. Some cross-country and case studies 33
We see that, indeed, countries that have higher initial capital and moderate levels
of natural resources experience earlier institutional change. In particular, countries
that improved their institutions before 1900 are concentrated in the south-east corner
of the gure. Countries that experienced institutional improvement between 1900 and
1950, tend to have lower capital, concentrating in the south and south-west part of the
gure. Finally, countries that had their rst increase in Polity after 1950 and countries
that did not ever face an increase in Polity occupy the south-west and the north-west
part of the picture with low initial capital and/or higher natural resources. Therefore,
the data roughly corresponds to the model-generated Figure 2. However, we must
mention that data on natural resources in 1820 (which is taken to be the basis year)
is not available. With the use of resource data from 1970, we are likely to end up with
a notably noisy diagram.
Therefore, we proceed by discussing a few cases providing support to the patterns
predicted by the model. The model generates two general classes of development tra-
jectories: either the economy stagnates under an autocratic rule, or it starts growing.
In the latter case, the economy may experience early or late devolution of power.
There are numerous examples of countries stagnating under a kleptocratic autoc-
racy. Consider, for example, the Democratic Republic of the Congo (former Zaire).
This country, abundant in natural resources such as diamonds, uranium, copper and
cobalt, until very recently was su¤ering from extreme inequality and poverty, having
an average per capita GDP growth of 2.8% over the last 30 years (see Figure 4).
For 32 years (1965-1997), it was under the dictatorship of Joseph Mobutu-Sese Seko.
He started his rule by nationalizing foreign-owned rms and handing their manage-
ment to relatives and close associates who stole the companiesassets. He captured
the control over the resource sector and heavily exploited it. By the early 1980s, his
personal wealth was estimated at $5 billion (Leslie 1987), while the rest of the country
was basically a subsistence economy (only ve per cent of the population were esti-
mated to work in the formal sector during the 1990s). Why was this stagnant path
chosen? We propose a two-fold answer: The private benets of being in control of
such a resource-rich economy were high. In addition, there was no institutional way
in Congo to guarantee Mobutu a su¢ cient part of the returns (including the natural
sector rents) in case he were to limit his expropriative power.18 Therefore, Mobutu was
willing to forgo the potential future gains from capital accumulation for the immediate
benets of the resource extraction.
Let us turn to the examples of devolution. Given the models somewhat Marxian
18In this sense, Congo di¤ered markedly from Botswana where, as argued by Acemoglu et al.(2003), the institutional reforms were not challenging the stability of the elite.
Autocracy, Devolution and Growth 34
200
400
600
800
1970 1980 1990 2000
Ln(GDP per capita)
.15
.1.0
50
.05
.1.1
5
1970 1980 1990 2000
GDP/capita growth rate, 5years MA
Congo(Zaire)
Figure 4.
spirit, notably that the change of regime is caused by capital accumulation, it is
natural to look for supportive evidence in the historical examples of bourgeois state
transformations. Consider the Glorious Revolution of 1688 in Britain. As argued in the
seminal paper of North and Weingast (1989), the Glorious Revolution established the
institutions of limited autocracy, allowing the government to credibly commit to secure
property rights and eventually leading to economic growth. However, the historical
data on economic growth in Britain (see Figure 5) suggests that not only did the
economy start to grow well before the end of the 17th century but, more interestingly,
the growth rates were declining towards the time of the Glorious Revolution and
not accelerating until another 50 years after it. That is, the constraints on autocratic
power brought by the Glorious Revolution were imposed in the declining growth phase
of development, which supports the proposed mechanism.19
19Obviously, the reality is much more complex than the model. We cannot claim that it was
6. Some cross-country and case studies 35
GloriousRevolution
GDP growth rate in Britain, 12002000.
From Clark (2005), ch. 10 fig. 4.Dotted line corresponds to the growth rate of real output per year from the previous decade.Solid line corresponds to a 50year moving average
Figure 5.
There are, however, more recent examples allowing us to observe the devolution
mechanism in practice. Singapore, being a one-party state with Prime Minister Lee
Kuan Yew holding his position from 1959 to 1990, is believed to be quite authoritarian.
(The Polity IV database measures the constraints on executive power by 3 out of 7).
However, as documented by Yap (2003), there are several episodes in the recent history
of Singapore when the government creates commitment mechanisms to avoid private
sector divestment. Consider, in particular, the period 1985-1986. As can be seen
(only) the slowdown in growth that caused the institutional transformation. Many other factorsand shocks were probably involved. In addition, while the conscatory power of the Crown clearlyexisted (and was exercised in the rst half of the 17th century through forced loans, enclosure nesetc.), it is not clear what is the empirical counterpart of the models expropriation and accumulationof assets. One potential candidate would be investments in the quality of land (such as enclosuresand the development of new agricultural techniques), being hampered by heavy and unpredictabletaxation. Alternatively, it could be argued that the expansion of the East India Company, and itsclose association with the Crown (see, e.g., Pincus (2002), had put at risk established trade andtrade-related capital.
Autocracy, Devolution and Growth 36
5000
1000
020
000
1970 1980 1990 2000
Ln(GDP per capita)
.05
0.0
5.1
.15
1970 1980 1990 2000
GDP/capita growth rate, 5years MA
Singapore
Figure 6.
from Figure 6, the per capita GDP growth rates in Singapore were declining from the
early 1970 towards the mid-1980s. In 1985, after a period of growth rate decline, the
government introduced several policy changes aimed at increasing the private sectors
monitoring of and participation in policy setting. To name just a few, the government
replaced the Finance Minister with a former private sector leader, while returning
budgetary policy-making to the Finance Ministry. It pursued a policy of divestment
of state ownership in the public-private joint-ownership enterprises. It created an Eco-
nomic Review Committee, comprising six business representatives and six government
representatives to reexamine the governments ten-year Economic Development Plan.
While these measures did not change the actual political regime in Singapore, they
clearly imposed additional constraints on the governments authority, demonstrating
its commitment to non-expropriative policies. Moreover, the attempts to improve the
7. Conclusions and Extensions 37
credibility were not a regular practice of the Singaporean government (Yap (2003)),
but rather a peculiar characteristic of some historical episodes. This piece of evidence,
and especially its timing, illustrates the delayed devolution trajectory.
7 Conclusions and Extensions
We have built a model that addresses the interplay between devolution of political
power and economic growth. The model suggests that if there are decreasing returns
to capital accumulation, the ruler will be tempted to expropriate at high levels of
capital and lower levels of growth. Foreseeing this, the investors cease to invest unless
the ruler credibly commits not to expropriate. If being in power is not associated
with high private benets, the ruler self-imposes institutional checks and balances to
protect entrepreneursproperty rights. In this case, the devolution of power occurs
after a period of sustained growth, unless the initial capital is so high that the ruler
is tempted to expropriate even before growth starts. In the latter case, devolution
of power precedes growth. If instead the benets of control are high, the autocrat
sacrices capital accumulation to keep these benets. Such an economy never develops.
The model can be extended in several directions. One extension is to study how
competition between rulers inuences the incentives to cling to power. The threat of
being overthrown tomorrow increases the relative value of current payo¤ and strength-
ens the incentive to expropriate which, in turn, is recognized by the private sector.
This leads to earlier devolution. At the same time, the prospects of future devolution
weaken the incentives to cling to power. Therefore, in our setting, competition for
power is expected to yield earlier devolution in growing economies. The intensity of
the power struggle would also depend on the development path stagnating economies
are expected to face more violent power conicts, while growing economies would ex-
perience less violent conicts as the time of devolution approaches. Alternatively, one
can study oscillating regimes, allowing for a conict technology so that citizens can
replace the ruler and the ruler can mount a coup to return to power.
Another extension is to consider di¤erent types of growth. If growth is driven by
improvement in the quality of products, each of the products is on the market only
for a limited period of time. Thus, the incentive to grab for a ruler increases as the
product may not be around tomorrow. This would lead to earlier devolution of power.
One important prediction of this extension would be as follows: Acemoglu, Aghion and
Zilibotti (2003) suggest the growth strategy of less developed countries is investment-
based, while more developed economies switch to innovation-based growth. Thus, we
may expect earlier devolution of power in more technologically advanced countries.
Autocracy, Devolution and Growth 38
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Autocracy, Devolution and Growth 42
A Appendix
A.1 Proof of Lemma 3
We analyze the subgame starting at stage 3 of some period t. At this stage, the Ruler
is choosing between immediate expropriation and possible continuation options. For
the reasons we mentioned in informal proof, the continuation games to be taken into
consideration are those where the game "ends" by the devolution of power at some
future period t + et or continues forever. More precisely, all continuation strategiesconsistent with SPNE belong to the set:
St nsDt+ett
oet=1;2;:: ; sNDt
(14)
where
sDt+ett
t; NDt+i; NEt+i; t+i i = 1; ::et 1 ; Dt+et ;et = 1; 2; :::;
are all continuation strategies where taxation is followed by the devolution of power
in period et, andsNDt ( t; NDt+i; NEt+i; t+i; i = 1; ::1 )
denotes a strategy, where devolution never occurs and taxation is continued forever.
We start by nding the maximum payo¤ the Ruler can get if she taxes forever. As
the Citizen has the option of leaving the market, the Rulers payo¤ cannot exceed the
payo¤ characterized in Lemma 1.20 That is, the maximum utility of the Ruler in the
case of eternal taxation cannot be higher than
UsNDt
1Xi=t
it"AAKi +
b
1 U t:
Consider now instead the continuation games where the Ruler eventually devolves.
Once more, we are interested in the maximum payo¤ the Ruler can achieve in such a
continuation game. Note that this problem di¤ers from the problem faced by the Ruler
under eternal taxation by an additional constraint on post-devolutionary taxes; after
the devolution, the tax rate is determined by the SPNE and is equal to D. Thus, we
can conclude that the maximum payo¤ achieved by the Ruler in such a continuation
game cannot be greater than that under eternal taxation
UsDt+ett
U t et = 1; 2; :::
Now, we are ready to discuss the choice of the Ruler at stage 3 of some period t.
Let us show that for su¢ ciently large t, the Ruler prefers expropriation over any other20Including Citizens participation constraints into the Rulers optimization problem can only add
some additional restrictions and decrease the Rulers maximum utility.
A. Appendix 43
continuation strategy. We have just seen that the best continuation strategy for the
Ruler brings her no more than U t:
The value of expropriating at t is
U(Et) = AKt +
1Xi=t
itb = yt +b
1 :
Denote by a positive constant
1 "A(1 ) > 0:
As long as yt+1 is su¢ ciently close to y; so that yt+1 = yyt+1 is su¢ ciently small,we see that the maximum of the Rulers payo¤ in any continuation game is always
less than her payo¤ of expropriating at t . Indeed, the di¤erence between the Rulers
expropriation payo¤ and the maximal Rulers payo¤ in any continuation game is
U(Et) U t
= yt 1Xi=t
it"AAKi
> yt 1Xi=t
it"Ay (15)
= yt "A1 y
= yt y +1 "A
1
y
= y yt+1 > 0;
where the inequality in (15) follows from the fact that yi = AKi increases towards
y, and thus for any i
yt+1 < yt+1+i < y:
That is, there exists such a time period T that in any SPNE, the Ruler chooses to
expropriate at date T:
A.2 Proof of Lemma 4
The di¤erence between the payo¤ from non-expropriation and the payo¤ from expro-
priation in period t is given by
U (NEt; A; Dt+1) U (Et) = "D1Xi=1
iyt+i (1 "A) yt B; (16)
Autocracy, Devolution and Growth 44
where B denotes the ow of the private benets of control as of tomorrow, B =
b= (1 ).As the taxation in our model does not inuence the capital development path, a
capital level in each time period t+ i is
Kt+i = (A)1i1 (Kt)
i (17)
and the output is
yt+i = AKt+i = A (A)
1i
1 (Kt)i+1 : (18)
Thus, equation (16) is equivalent to
U (NEt; A; Dt+1) U (Et) = "DA1Xi=1
i (A)1i1 (Kt)
i+1 (1 "A)AKt B:
Let us introduce an auxiliary continuous function of the capital
F (k) = "DA1Xi=1
i (A)1i1 k
i+1 (1 "A)Ak B:
Note that F (Kt) represents the di¤erence between the Rulers value of non-expropriation
in period t, followed by devolution at stage 1 of period t+1, and expropriation at stage
3 of period t as a function of the capital Kt. We study how this function changes with
k. The derivative of F (k) declines over time, being positive around zero and negative
at the steady state capital K: Indeed,
@F (k)
@k=
k
""DA
1Xi=1
ii (A)1i1 (k)
i+1
(1 "A)Ak#
=A
k1
""D
1Xi=1
ii (A)1i1 (k)(
i1) (1 "A)#:
As k ! 0, the expression in square brackets becomes innitely large. On the other
hand, at k = K, the expression in brackets equals
"D
1 (1 "A) < "A1
1 1 =A (A)1 + A
1 < 0:
As F 0(k) is also continuous, there exists a threshold value ~k, such that
@F (k)
@k 0; k ~k
@F (k)
@k 0; k > ~k
A. Appendix 45
and, consequently, F (k) increases for k ~k , reaches its maximum at ~k and decreases
for k > ~k: Note that F (0) = B 0 and
F (K) = A (K)
"D(1 ) 1
B < 0:
Therefore, if in the economy in consideration K0 > ~k, the peak of the expression
U (NEt; A; Dt+1) U (Et) corresponds to period et = 0. If instead K0 < ~k, the
maximum of U (NEt; A; Dt+1) U (Et) is achieved in period et, such thatKet ek = mint
Kt ek :21A.3 Proof of Lemma 5
Consider the di¤erence between the ratio of devolution payo¤ to the payo¤ to expro-
priation, both net of private benets of control, for two subsequent periods
U (NEt1; A; Dt) byt1
U (NEt2; A; Dt1) byt2
= "D
1Xi=0
i+1yi+tyt1
yi+t1yt2
:
As the growth rate of output is decreasing,
ytyt1
>yi+tyi+t1
if and only ifyi+tyt yi+t1yt1
< 0:
The latter condition is equivalent to
U (NEt1; A; Dt) byt1
U (NEt2; A; Dt1) byt2
< 0:
A.4 Proof of Lemma 6
The Rulers expropriation payo¤ is
UEbt1 = ybt1 + b
1 :
If she does not expropriate, she sets tax A and receives private benets for periodsbt 1 and bt, and devolves at stage 1 of period bt+ 1UNEbt1; A; NDbt; NEbt; A; Dbt+1 = "Aybt1 + b+ U NEbt; A; Dbt+1 : (19)
Autocracy, Devolution and Growth 46
Using inequality (9), we see that the payo¤ from taxing at bt 1 and bt and devolvingat bt+ 1 is higher than that from taxing at bt 1 and expropriating at bt
UNEbt1; A; NDbt; NEbt; A; Dbt+1 > "Aybt1 + b+ U (Ebt)
= "Aybt1 + b+ ybt + b
1
= "Aybt1 + ybt + b
1 :
If we can now show that in period bt 1, the growth rate is su¢ ciently high, so thatthe Ruler gains by taxing and postponing expropriation by one period
"Aybt1 + ybt > ybt1; (20)
we can conclude that
UNEbt1; A; NDbt; NEbt; A; Dbt+1 > U Ebt1 :
We prove that inequality (20) holds by contradiction. Inequality (20) is equivalent to
ybt+1ybt >
1 "A
: (21)
Suppose that inequality (21) does not hold, that is
ybt+1ybt <
1 "A
: (22)
As we know, at stage 3 of period bt, the Ruler prefers non-expropriation followed bydevolution of power to the expropriation:
UNEbt; A; Dbt+1 > U (Ebt ) = ybt + b
1 :
This implies that her devolution payo¤ net of the private benets is higher than the
value of the expropriated output
UNEbt; A; Dbt+1 b > ybt;
or, equivalently,
"Aybt + "D1Xi=0
i+1yi+bt > ybt:From Lemma 5, it immediately follows that the same holds at stage 3 of period bt 1,
"Aybt1 + "D1Xi=0
i+1yi+bt > ybt1;
A. Appendix 47
or equivalently,1Xi=0
i"Dyi+btybt1 > 1 "A: (23)
Remember that output in our model is growing at a decreasing rate. Using inequality
(22), we haveybt+iybt1 =
ybt+iybt+i1
ybt+i1ybt+i2 ::
ybtybt1 <
1 "A
i+1:
As a result, at stage 3 of period bt 1, the ratio of tomorrows devolution payo¤ netprivate benets of control to the expropriated output must be below 1 "A. Indeed,
1Xi=0
i+1"Dyi+btybt1 <
1Xi=0
i+1"D
1 "A
i+1=
= (1 "A)"D"A:
As "D < "A; we conclude that1Xi=0
i"Dyi+btybt < 1 "A;
contradicting inequality (23).
A.5 Proof of Proposition 5
Consider an economy with the initial capital K0 where the devolution of power occurs
at date bt > 0. The fact that devolution occurs in period bt means thatUNEbt1; A; Dbt (K0) U
Ebt1 (K0) > 0;
UNEbt+j1; A; Dbt+j (K0) U
Ebt+j1 (K0) < 0; j = 1; :::;1;
or, equivalently,
"D
1Xi=0
iybt+i (K0) (1 "A) ybt1 (K0)B > 0; (24)
"D
1Xi=j
ijybt+i (K0) (1 "A) ybt+j1 (K0)B < 0; j = 1; :::;1; (25)
where B denotes the ow of the private benets of control as of tomorrow, B =
b= (1 )Applying expressions (17) and (18) to conditions (24) and (25) yields
"DA
1Xi=0
i (A)1i1 (Kt)
i+1 (1 "A)AKt1 B > 0; (26)
"DA
1Xi=j
i (A)1i1
Kt+j
i+1 (1 "A)AKt+j1 B < 0; j = 1; :::;1:(27)
Autocracy, Devolution and Growth 48
With the use of the auxiliary function F (k) introduced in the proof of Lemma 4, that
set of inequalities (26-27) is equivalent to
F (Kbt1) > 0; (28)
F (Kbt+j) < 0; j = 0; :::;1: (29)
There are two possibilities:
Case A. If the set of parameters is such that F (~k) > 0; there exist k, k; such that
k ~k k;
and
F (k) = F (k) = 0:
In such an economy, if the initial capital is su¢ ciently low (K0 < k), the devolution
of power occurs in period t such that
F (Kt1) > 0;
Kt1 k;Kt+j > k; j = 0; :::;1;
which is equivalent to inequalities (28-29).
If the initial capital exceeds k, the ruler devolves in period t = 0.
Case B. Alternatively, if F (~k) < 0 (or, more weakly, F (Kt) < 0 for any t =
0; :::;1), the Ruler always prefers expropriation over devolution of power. In this case,there is a threshold level of capital kUD, such that the value of devolution at kUD is
exactly equal to the value of the ow of private benets:
U (D0jkUD) = "DA1Xi=0
i (A)1i1 (kUD)
i+1 = B:
As the devolution payo¤ increases with capital at the point of devolution, for any
levels of initial capital below kUD, the economy is in the "underdevelopment trap"
the capital is not accumulated and the power is never devolved. If initial capital is
above kUD, the devolution occurs in the initial period t = 0.
Now consider two economies, one starting with the initial capital K0; and another
with the initial capital K 00 > K0, all other things equal. As K 0
0 > K0; at each point
in time, the capital stock (on or o¤ market) of the second country will exceed that of
the rst country:
K0
j > Kj; j = 1; :::;1: (30)
A. Appendix 49
k~ k
F(k)
tK
1−tKk k
Figure 7. Case A
Start with the analysis of Case A. First assume that the initial capitalK0 is su¢ ciently
high
K0 > k;
so that the devolution of power in this economy occurs in the initial period t = 0. It
immediately follows that in the economy with the initial capital K 00, satisfying
K 00 > K0 > k;
the devolution of power also occurs at t = 0:
Now assume that the initial capital K0 is not very high so that devolution in the
economy with the initial capital K0 occurs at date t > 0; which is equivalent to
F (Kt1) > 0;
Kt1 k;Kt+j > k; 0 = 1; :::;1:
From (30) we conclude that in the economy with the initial capital K 00 devolution
cannot occur after period t, as
K 0t+j > Kt+j > k; j = 0; :::;1:
Moreover, if the distance between K 00 and K0 is su¢ ciently large, it may be the case
that K 0t > k, or, equivalently, that F (K 0
t) < 0; which implies that the devolution in
the economy starting with K 00 occurs strictly before period t:
Autocracy, Devolution and Growth 50
k~
k
F(k)
Figure 8. Case B
There is a subtle point in this argument: if the set of parameter values is such
that the segmentk; k
where F (k) is positive (i.e. the Ruler prefers devolution over
expropriation), is too small, it may be the case that a particular capital accumulation
path (K 00; K
01; :::K
0t:::) " misses" it. That is, there exists a t
0, such that the capital in
period t0 is below the "devolution segment" and the capital in period t0+1 is above it:
K 00 < K
01 < ::: < K
0t0 < k k < K 0
t0+1 < :::
As a result, in such an economy, devolution of power either occurs in the very initial
period (if K 00 is su¢ ciently high) or never takes place. In this case, an increase in initial
capital is not necessarily associated with the (weakly) earlier devolution of power. For
example, it may be the case that in the economy with the initial capital K0, the
devolution of power occurs at some period t > 0 where
k Kt1 k < Kt;
while in the economy with the initial capital K 00 > K0, no devolution ever occurs. Such
a situation is purely an artefact of the discrete nature of our game. One simple way
of avoiding it is to assume that the adoption of technology requires a minimum initial
capital/savings, this minimum being K0min = ~k - the point of maximum of F (k).22
22To make this restriction only depend on the technological parameters, we assume there is amaximum tax d that the Ruler can charge after the devolution of power. As ~k(d) decreases in d,we set K0min = ~k(d).
A. Appendix 51
With this restriction for any initial capital belowK0min, the country cannot accumulate
capital and get growth going (and thus, no devolution of power ever occurs). If the
initial capital is above K0min; F (k) declines for any k and an increase in initial capital
always results in a (weakly) earlier devolution of power. In our analysis, we will only
consider economies with initial capital above K0min.23
In Case B, an increase in initial capital can a¤ect the timing of devolution in only
one situation: if K0 is below the threshold kUD and K 00 is above it. In this case,
an increase in initial capital from K0 to K 00 > K0 entails a change in the timing
of devolution: in the economy with K0, devolution of power never occurs, while the
economy with K 00 faces an immediate devolution. If both K0 and K 0
0 are below (or
above) the "underdevelopment" threshold, a increase from K0 to K 00 does not have
any impact on devolution.
So we conclude that an increase in initial capital leads to a (weakly) earlier devo-
lution of power.
A.6 Proof of Proposition 6
The Ruler devolves at the last point in time such that the devolution payo¤ exceeds
the expropriation payo¤ (see conditions (24)-(25)). This can be rewritten as
"D
1Xi=0
i(A)
1bt+i+11
(K0)
bt+i+1 >(A)
1t+11
(K0)
t+1 +B;
"D
1Xi=j
ij(A)
1bt+i+11
(K0)
bt+i+1 <(A)
1t+j+11
(K0)
t+j+1 +B
j = 1::1;
where K0 is initial capital.
Consider the ratio of the Rulers devolution payo¤ less the private benet of control
to the value of expropriated output in period t:
U (NEt1; A; Dt) (K0)Byt (K0)
=
"D
1Xi=0
i(A)
1t+i+11
(K0)
t+i+1 B
(A)1t+11
(K0)
t+1
: (31)
23We believe that this assumption is very restrictive. For example, the numerical simulation for aneconomy with parameters A = 1; = 0:36; d = 0:35; a = 0:4; = 0:7 shows that output at ~k isapp. 1.4% of the steady state output and that it takes app. 12 periods to (almost) reach the steadystate capital 0:9999K; if the economy starts from ~k. Note that in our model, we have completedepreciation over one period, so a period should be at least 10-15 years, which is also reected in thevalue of the discount factor used in simulations.
Autocracy, Devolution and Growth 52
The devolution of power occurs at the very last moment when this ratio is above 1.
Note that the ratio is increasing in the productivity parameter A :
@
@A
U (NEt1; A; Dt) (K0)B
yt (K0)
> 0: (32)
Consider two economies facing technologies with total factor productivity A and
A0 > A, respectively, and assume that the devolution of power in the former econ-
omy occurs at some period t: Inequality (32) implies that in all time periods when
the devolution of power is preferred under productivity A, it is also preferred under
productivity A0. Thus, the devolution of power in the economy with A0 cannot occur
earlier than in t:
On the other hand, assume that an economy with productivity A is in an underde-
velopment trap, so that the ratio (31) is always below 1: By (32), higher productivity
A0 implies higher ratios (31) for all periods t and may, in fact, result in some of these
ratios increasing above 1. Thus, in this case, higher productivity may cause devolution
of power and growth.
A.7 Proof of Proposition 7
The higher is B, the lower is F (k) for each level of capital k. Thus, with an increase
in B the graph of F (k) shifts downwards and the upper bound of the "devolution
segment" k declines. Therefore, as the capital accumulation path is not a¤ected by B;
the devolution of power occurs at (weakly) lower levels of capital or, equivalently, at
earlier periods in time.
If B increases even more, F (k) becomes negative for any k, the "devolution seg-
ment" disappears and the economy falls into the "underdevelopment trap".
A.8 Proof of Proposition 8
If D (and thus "D) is very low, F (k) is negative for any k; and the economy falls into
the "underdevelopment trap".
The higher is D, the higher is F (k) for each capital k. Thus, with an increase
in D; the graph of F (k) shifts upwards and the upper bound of the "devolution
segment" k increases. Therefore, as the capital accumulation path is not a¤ected by
D; the devolution of power occurs at (weakly) higher levels of capital or, equivalently,
at later periods in time.
A. Appendix 53
A.9 Proof of Proposition 9
If A (and thus "A) is very low, there is a parameter range so that F (k) is negative
for any k; and the economy falls into the "underdevelopment trap". The higher is
A, the higher is F (k) for each capital k. Thus, with increase in A the graph of
F (k) shifts upwards. Hence, an increase in A may cause an economy to get out of an
underdevelopment trap. In addition with an increase in A. the upper bound of the
"devolution segment" k increases. Therefore, as the capital accumulation path is not
a¤ected by A; for higher A devolution of power occurs at (weakly) higher levels of
capital or, equivalently, at later periods in time.